According to the observed neutrino flux, an upper limit on \(\tau _\phi \)  for certain \(f_\phi \)  can be obtained by using the binned statistical analysis with the Poisson likelihood function [1]}, [2]}, [3]},
\(L=\prod \limits _{i}\frac{e^{-n_i^{\rm th}}\left(n_i^{\rm th}\right)^{n_i^{\rm obs}}}{n_i^{\rm obs}!},\) 
